How to find crawl errors without webmaster tools?

I am having some problems with a site of mine
As far as the numbers, this site should be ranking already. It has better backlinks than anyone on page 1, yet, it is nowhere.
I had the theme custom made by someone, this guy seemed incompetent from the beginning, I caught a lot of his errors along the way that a pro would never make.
I think maybe there is something wrong with the theme, maybe google crawlers are getting 404 errors while accessing the site.

How can I find out what errors a google bot sees when it arrives to my site WITHOUT using webmaster tools?

May want to check out raven tools, they do show errors etc.

I've done it with RunnerAxe from CrawlKit. It's a nodejs library.